Particles on Deforming Surfaces: Theory & Analysis

[andresB]

All books in analytical mechanics explain the case of a particle moving on a given static surface. But what happen if, for example, the surface is having some deformation?. I imagine that the principle of virtual work, and hence, D'Alembert are no longer valid since the normal force by the surface do work. Hence Hamilton principle no longer work either.

is there a theory for a particle moving on such deforming surface?

 

[FactChecker]

Can you specify the force is that is deforming the constraint? That is the source of the work.

 

[andresB]

Can you specify the force is that is deforming the constraint? That is the source of the work.

Well, It doesn't matter I think.

Mathematically, think in point moving along the surface S(x,y,z,t)=0, where the surface dependence on time is given beforehand and it is not affected by the motion of the particle.

 

[wrobel]

By definition ideal constraints do not work on virtual displacements in your case $\delta x,\delta y,\delta z,$
$$\frac{\partial S}{\partial x}\delta x+\frac{\partial S}{\partial y}\delta y+\frac{\partial S}{\partial z}\delta z=0$$
No problem, the D'Alembert-Lagrange principle keeps holding as well as all the Lagrangian formalism
For details see D T Greenwood Classical Dynamics

 

[andresB]

For details see D T Greenwood Classical Dynamics

That was a really good read, thank you.